Imitation versus payoff - duality of the decision-making process demonstrates
criticality and consensus formation
M. Turalska1 and B.J. West1,2
1
arXiv:1409.3162v1 [nlin.AO] 10 Sep 2014
2
Physics Department, Duke University, Durham, NC 27709, USA
Information Science Directorate, US Army Research Office, Research Triangle Park, NC 27708, USA
(Dated: September 11, 2014)
We consider a dual model of decision making, in which an individual forms its opinion based on
contrasting mechanisms of imitation and rational calculation. The decision making model (DMM)
implements imitating behavior by means of a network of coupled two-state master equations that
undergoes a phase transition at a critical value of a control parameter. The evolutionary spatial game
(EGM), being a generalization of the Prisoner’s dilemma game, is used to determine in objective
fashion the cooperative or anti-cooperative strategy adopted by individuals. Interactions between
two sources of dynamics increases the domain of initial states attracted to phase transition dynamics
beyond that of the DMM network in isolation. Additionally, on average the influence of the DMM
on the game increases the final observed fraction of cooperators in the system.
PACS numbers: 02.50.Le, 89.75.Da, 02.50.Ey, 05.50.+q
I.
INTRODUCTION
In a society interconnected by family ties, friendships,
acquaintances or work relations it is unavoidable that
a person’s behavior or decisions depends on the choices
made by other people. The surrounding social network
influences the opinions we hold, the products we buy or
the activities we pursue. Therefore exploring the basic
principles that give rise to social processes in which individual behavior aggregates into collective outcomes can
provide significant insight into the individual’s decisionmaking process.
Human performance with regard to decision-making
can be viewed from at least two complementary perspectives. One is from the assumptions made about how individuals behave and to predict the outcomes of their
interactions at a group level. This outlines a mechanistic
approach to the study of decision-making. The predictions based on this approach are tested by comparing
them with observational data of how individuals behave
in response to other individuals and to the environment.
In this way the mechanisms through which collective behavior is generated are determined. At the turn of the
twentieth century Tarde [1] argued that imitation was
the fundamental mechanism by which the phenomena of
crowds, fads, fashions and crime, as well as other collective behaviors, could be understood. At the same time,
Baldwin [2] maintained that the behavior based on imitation arose out of the mental development of the child
resulting from imitation being a basic form of learning.
Currently, imitation remains an important concept in
the social sciences, being pointed to as a mechanism responsible for herding, information cascades [3] or many
homophily-based behaviors [4]. Social experiments such
as the Friends and Family study [5] demonstrated higher
efficiency of incentives directed at the social network of
an individual rather than directly offered to a person,
suggesting strong influence that the actions of our peers
have on our own decisions [6]. Similarly, coping proved
to be the preferred and most effective strategy to acquire
adaptive behavior in complex environment [7], even when
other non-social sources of information were available at
the same cost.
This mechanistic approach to decision-making is separate and distinct from the functional approach, in which
we ask what is the value or function of a particular behavioral strategy. The basic assumption of the second
approach is that a given behavior can be rationally evaluated in terms of costs and benefits, which allows for
an objective comparison of alternative strategies. This
principle of balancing costs against benefits to arrive at
a decision is central to modern economical [8], political
[9] and social science [10].
In mathematical terms the functional approach aligns
with the basic assumptions of game theory, which originated from games of chance. Game theory influenced
behavioral sciences through the introduction of the utility function by Daniel Bernoulli in 1730 [11]. In doing
so Bernoulli resolved the famous St. Petersburg paradox
[12], demonstrating that a rational strategy should be
based on the subjective desirability of a game’s outcome
rather than being proportional to the game’s expected
value. The suggestion that the value of a thing to an
individual is not simply equivalent to its monetary value
reached its full articulation in the voices of von Neumann
and Morgenstern [13] in their seminal work on game theory and economics.
More recently game theory was used to study the emergence of cooperative behaviors, as a way of obtaining insight into this evolutionary puzzling phenomena. The
work of Nowak and May [14] for the first time extended
game theory principles to spatial networks, and demonstrated that the introduction of spatial structure between
players lead to spatially and temporally rich dynamics. Following this observation, the impact of the spatial
structure on the evolution of cooperation has been investigated in detail [15–19]. Contrary to the well-mixed
case, where non-cooperative behavior is favored, the well-
2
known Prisoner’s Dilemma game performed on a square
lattice with next-nearest-neighbor interactions promotes
cooperation. In the effort to investigate the impact of
different interaction topologies, heterogenous topologies
were investigated more recently, with scale-free architecture being the most extensively studied [20, 21].
Herein we explore both mechanistic and functional perspectives as a way to understand the decision-making
process. We consider a society to be given by a twolayer network, whose elements are individuals making decisions simultaneously using two distinct sets of criteria,
as indicated in Fig. 1. On the one hand, individuals form
their decisions based on the perception of actions and appearances of their neighbors, adopting the concept of imitation. This behavior, captured by the decision making
model (DMM) [22, 23] demonstrates the cooperative behavior induced by the critical dynamics associated with
a phase transition. This behavior is counterbalanced by
the rationality of the evolutionary game model (EGM).
Thus, while personal decisions are influenced by the desire to be liked and accepted, individuals also weigh the
effect of certain potential relations on their careers, balancing the cost and payoffs of such relations. This latter behavior is captured by the rational and deterministic EGM rules in the spirit of the original approach
of Nowak and May [14]. Thus, the adopted two-layer
network model allows us to refer to two aspects of the
decision-making process of a single individual, each aspect being defined by distinct dynamic rules. The two
layers interact with one another and modify their separate dynamics.
In Section II we review the basic properties of the
DMM network in isolation, that is, not interacting with
the second network. The phase transition properties of
the DMM network for different interaction strengths are
briefly discussed. In Section III we outline the basic properties of the EGM network in isolation. In this case the
traditional game theory final states are identified. Finally, in Section IV the two functionally distinct networks are allowed to interact with one another and the
difference in the asymptotic states due to their mutual
interaction are analyzed and discussed. We draw some
conclusions in Section V.
II.
ISOLATED DMM LAYER
In the DMM network the state of an isolated individual
si (t) is described by a two-state master equation,
dp(i) (t)
= G(i) (t)p(i) (t),
dt
where G(i) (t) is a 2 × 2 transition matrix:
(i)
(i)
−g+−
(t) g−+
(t)
G(i) (t) =
(i)
(i)
−g−+
(t) g+−
(t)
(1)
(2)
and the probability of being in one of two states (+1, −1),
(i) (i)
is p(i) (t) = (p+ , p− ). Positioning N such individuals at
FIG. 1: (Color online) Decision-making process in a society
presented as a two-layer network. The lower layer models
the imitation behavior, with positive (+1) and negative (-1)
opinions held by an individual of the decision making model
being represented by black and white nodes, respectively. The
upper layer models the rational behavior of the evolutionary
game, with nodes selecting between two strategies: being a
protagonist (P, blue circle) or antagonist (A, red circle). The
color of the arrows in the lower layer (P, protagonist - red;
A, antagonist - blue) indicates that strategy adopted in the
upper layer influences the character of the imitating behavior.
the nodes of a square lattice yields a system of N coupled two-state master equations [22, 24] which, under
the assumption of nearest neighbor interactions, contain
time-dependent transition rates for each of the i individuals:
K
(i)
g+−
(t) = g0 exp
(M+ (i, t) − M− (i, t)) ;
M
K
(i)
(3)
g−+
(t) = g0 exp − (M+ (i, t) − M− (i, t))
M
where K is the strength of the interaction. On the twodimensional lattice M = 4 and 0 ≤ M± (i, t) ≤ 4 denotes
the count of nearest neighbors in states ±1 at time t.
(i)
In the mean field approximation M± (i, t)/M → p± (t)
and the transition rates become exponentially dependent
on the state probabilities resulting in a highly nonlinear
master equation [25].
A wealth of results exist for the dynamics of DMM
on networks of various topologies, in a configuration of a
single network, as well as in the case of coupled networks
[25]. Here we concentrate on the global behavior of the
3
FIG. 2:
(Color online) The phase transition property of
a predominantly protagonist DMM lattice is indicated with
an increasing number of antagonists. The black dots denote
an all protagonist lattice, the dots receeding from the first
calculation depict 1%, 3%, 5%, and the squares depict 10%,
15% and 20% antagonists in sequence.
model, which is defined by the fluctuations of the global
variable
N
ξ (K, t) =
1X
si (K, t)
N i=1
(4)
which is further used to calculate the equilibrium global
value ξeq ≡ h|ξ (K, t)|i . When the coupling parameter
K > 0, single units of the system become more and more
cooperative and for coupling value larger than a critical
one, K > KC , the interaction between units is strong
enough to give rise to a majority state, during which a
significant number of nodes adopts the same opinion at
the same time. Thus, the global dynamics of the DMM
is characterized by a phase transition with respect to the
coupling parameter K (Fig. 2), demonstrating that a
system of identical units imitating each other’s actions is
able to reach consensus, given sufficient influence of the
imitation on their decisions.
However, this model society in which all members are
interacting only through positive relationships (friendships, collaborations or sharing of information) is not
very realistic, since negative effects are also present in
most circumstances. Some relationships are friendly,
while others are antagonistic or even hostile and interactions between people often lead to disagreement and
conflict. Thus, there is a need to modify the basic DMM
to include a mix of positive and negative relationships.
Rather than define the nature of the relationship between two nodes, which is usually done by introducing
links with positive (reciprocal friendship) and negative
(mutual antagonism) signs [26], we consider a network in
which the nature of the interaction between individuals
depends on the individuals themselves. Thus, our modeled society is composed of two kinds of individuals: those
that always cooperate (protagonists) and those that always oppose the opinion of their peers (antagonists). As
a result we observe three kinds of interaction between two
nodes: reciprocal friendship and mutual antagonism, as
well as cooperator-antagonist pairs, in which one node
wants to cooperate while another opposes any kind of
mutual action.
On the one hand cooperating individuals operate according to the DMM dynamics defined in Eq. 3. On the
other hand, antagonists at any time oppose the opinion
of their neighbors, and their dynamics are defined by the
transition rates
K
(i)
g+− (t) =
g0 exp − (M+ (i, t) − M− (i, t)) ;
M
K
(i)
g−+ (t) =
(M+ (i, t) − M− (i, t))
(5)
g0 exp
M
where the only difference with respect to the dynamics of
protagonists is an opposite sign of the coupling constant.
All numerical calculations in this section are performed
on a square lattice of N = 20 × 20 nodes with periodic
boundary conditions. The initial state of each individual
is randomly assigned. In a single time step a computer
calculation involving the entire lattice is performed and
for every element si the transition rate of either Eq. 3 or
Eq. 5 is calculated, according to which element is given
the possibility to change its state. The transition rate for
a non-interacting unit is g0 = 0.01. The equilibrium value
ξeq is calculated as an average over 106 consecutive time
steps, after the same number of time steps since the initialization has passed thereby insuring that all transient
behavior has died away. The assignment of protagonist
and antagonist behavior is done randomly.
The dynamics of the DMM lattice with increasing
numbers of antagonists is depicted on Figure 2. It is evident from the figure that the phase transition the DMM
dynamics undergoes is sensitive to the fraction of the network members that are antagonists. The DMM lattice
dynamics undergoes a phase transition and this criticality persists with up to 5% antagonists randomly placed
on the lattice. However, the phase transition is clearly
suppressed when the number of antagonists is 10% and
above. These results are consistent with those determining the influence of committed minorities on group consensus [27, 28].
III.
ISOLATED EGM LAYER
The evolutionary game model (EGM) used herein is
a generalization of the Prisoners’ Dilemma (PD) game.
Traditionally the PD game consists of two players, each
of whom may choose to cooperate or defect in any single encounter. If both players choose to cooperate, both
4
FIG. 3: (Color online) The equilibrium fraction of protagonists present in the EGM lattice of N = 20 x 20 nodes is
plotted for the game parameters R = 1 and P = 0. The initial fraction of protagonists was 50%, randomly distributed
over the lattice. Periodic boundary conditions are considered. The region T > 1, S ≤ 0 locates the Prisoner’s Dilemma
(PD) game; the Stag Hunt (SH) game is found in the domain
0 ≤ T ≤ 1, S ≤ 0; the Snowdrift (SD) game is bounded by
T ≥ 1, 0 ≤ S ≤ 1; the Leader (LD) game is confined by T = 1
and to the right of the S = T diagonal; and the Battle of the
Sexes (SX) is above the diagonal S = T and bordered on the
left by T = 1.
get a reward pay-off of size R; if one defects and the
other cooperates the defector receives a ”temptation”
pay-off T while the cooperator receives a ”sucker’s” payoff S; if both defect, both receive ”punishment” pay-off
P.
The ordering of the payoff parameters given by
T > R > P > S defines the PD game. Nowak and
May [14] considered a two-dimensional lattice over which
the PD game was played in a sequential fashion, where
at each time step every node was able to change its strategy (defect or cooperate) depending on the outcome of
the game played with its neighbors at the previous time
step. The historical nomenclature of cooperator and defector is here replaced with protagonist and antagonist,
respectively, which we believe to be more compatible with
the language of the two-layer network model.
Consider the EGM network dynamics without it being
coupled to the DMM network dynamics. As depicted on
the top part of Figure 1, the protagonists and antagonists
are placed on the sites of a two-dimensional lattice and
interact only with their four nearest neighbors. In each
generation every individual plays adeterministic game
R S
defined by a pay-off matrix
with all its neighT P
bors. The pay-off gained by each individual at the end of
each generation is determined by summing payoffs of 2×2
games with each of its neighbors. The scores in the neighborhood, including the individual’s own score are ranked,
and in the following generation the individual adopts the
strategy of the most successful player from among its
neighbors. In the case of a tie between the scores of cooperative and antagonistic players, the individual keeps
its original strategy. Thus, adopted evolutionary strategy
is to act like the most successful neighbor.
Even this simple and completely deterministic situation leads to a wide array of behaviors. Figure 3 depicts
the equilibrium fraction of protagonists present in the
EGM game as a function of the two parameters S and
T . Without loss of generality, we assume that R > P
and normalize the pay-off values such that R = 1 and
P = 0. The initial configuration of the game consists of
50% protagonists and antagonists, distributed randomly
on the lattice.
It is evident that in the domain T ≤ 1, for almost
all values of S, that the equilibrium state is dominated
by protagonists. These regions are determined to have
between 5% and 10% randomly distributed antagonists
asymptotically. Whereas for T ≥ 1 and S < 0, the region
of the PD game, antagonists dominate with the network
having between 5% and 10% randomly distributed protagonists. The remaining regions have differing levels of
protagonists at equilibrium. The traditional games for
which there is a substantial literature are marked and
are not addressed here in more detail. We merely note
that selected ST parameter values enable us to determine
the outcome for the two-layer network and thereby determine the mutual influence of the DMM and EGM dynamics and the relative influences of imitation and payoff
on decision making in all these cases.
IV.
TWO-LAYER NETWORK
The constant struggle between maximizing individual
gains and the desire to be part of a community is modeled
by an interaction between EGM and DMM layers (Fig.
1). The coupling is realized dynamically, since the behavior of a node in the DMM layer depends on its strategy
in the EGM layer, being protagonist or antagonist. In
return the local configuration of nodes in the DMM layer
can change the strategy of a node in the EGM layer.
More precisely, each time step of the simulation consists of four operations:
1. One step of the EGM dynamics is realized in the
EGM layer. The pay-off for each individual is evaluated and used to update its strategy in the next
generation.
2. The following generation of protagonists and antagonists is used to define the sign of a coupling
constant K in the DMM layer. Consequently, a
node in the DMM layer may be a protagonist at
one time step, while in the next it acts as an antagonist, due to the changes made to the EGM layer.
3. One step of the DMM dynamics is performed in the
DMM layer, allowing nodes to change their state
5
FIG. 4: (Color online) The ratio D of the final fraction of
protagonists observed in the DMM-EGM system to the fraction of protagonists observed in the EGM network in isolation
is color coded. Grey area depicts the range of parameter values whose DMM-EGM network dynamics give rise to phase
transition. X depicts ST values for which fraction of protagonists in the isolated EGM network is smaller then 90%.
from ±1 to ∓1, or not change at all. This step
potentially affects the local neighborhood of an individual, when at one time step an individual is
surrounded by mostly other individuals of the same
sign, and in the next time step it is surrounded by
individuals of opposite sign.
4. A change to the DMM layer finally affects the strategy of an individual in the EGM layer. The decision
to change strategy in the EGM layer is made if the
average state of the local neighborhood of an individual in the DMM layer is of an opposite sign with
respect to that individual.
After those four steps, one full iteration of the twolayer network is completed, and the time index is advanced.
Figure 4 summarizes the asymptotic dynamics of the
two-layer network for a broad set of S and T values. The
color at each value of S and T is determined as the ratio
of the final fraction of protagonists observed in the DMMEGM network to the fraction of protagonists observed in
the EGM network in isolation:
ProtagonistsDM M −EGM
D = log10
(6)
ProtagonistsEGM
The large grey area depicts the range of parameter
values whose DMM-EGM network dynamics give rise to
phase transition. The initial fraction of 50% protagonists increases with the phase transition to a final state
of complete consensus among protagonists. The symbol
FIG. 5: (Color online) Typical equilibrium dynamical behavior for the global variable is depicted in four regions of (S,T)space. The blue dots denote the equilibrium fraction of protagonists and the red dots denote value of the global variable
in the two-layer system. The dashed line marks the equilibrium solution of the EGM in isolation. a) T ≤ 1, −1 ≤ S ≤ 3
global variable undergoes a phase transition at a Kc ≈ 2.2; b)
T ≥ 1, S ≤ 1 (narrow band of Fig.4 with no phase transition),
the level of protagonists is below that of EGM in isolation;
c) the predominantly green region in Fig. 3 with no phase
transition and the fraction of protagonists the same as for the
isolated EGM; d) the PD segment of Fig. 3 has no phase
transition and a low fraction of protagonists that is above the
isolated EGM level.
X depicts ST values for which the fraction of protagonists in the isolated EGM network is smaller then 90%
and consequently if the DMM were operating in isolation
on such a configuration of protagonists and antagonists,
the phase transition would be strongly suppressed. However the coupling of DMM-EGM networks facilitates the
critical transition at these values.
The remaining regions do not undergo phase transition, but do show a wide range of dynamic behaviors.
Four types of typical dynamics for the two-layer DMMEGM network can be determined. For each of the panels
in Figure 5 the DMM-EGM network was initialized with
50% antagonists randomly distributed. Such an initial
state is sufficient to suppress a phase transition for the
DMM in isolation (Fig. 2).
However in Figure 5a, as the control parameter K is increased, the average global variable of the DMM layer in
the DMM-EGM network manifests critical behavior. Simultaneously, the 10% antagonists seen in isolated EGM
layer change their strategy and are converted to protagonists in the DMM-EGM network. For reference the
dashed curve shows the fraction of protagonists in the
isolated EGM network.
We demonstrated in our earlier work [28] that the
global dynamics of the DMM belongs to the Ising universality class. The investigation of the scaling properties of the global order parameter and the susceptibility
confirms that property also for the dual-layer dynamics.
6
Despite the relatively small size of the system, a lattice
of N = 20 x 20 nodes, on which the DMM dynamics is
realized in the isolated layer configuration, the values of
the mean field and susceptibility in the vicinity of the
phase transition point scale as a power law (Fig. 6 top
row). When coupled to the EGM layer, the scaling behavior and exponents are preserved (Fig. 6), suggesting
that a very small fraction of antagonists present in the
system does not cause a significant change in dynamical
properties of the system.
Figure 5c depicts representative behavior of the DMMEGM dynamics for ST values from the upper right quadrant of Fig. 3: S, T > 1. Here the fraction of protagonists is typically the same as that of the isolated EGM
(the pale blue region of Fig. 4). Figure 5b is taken from
a narrow transition channel parallel to T = 1 in which
the fraction of protagonists in the dual network is typically below that of the isolated EGM network, indicating
a marked change in behavior for small increases in temptation. Across this channel the arrangement of the payoff matrix converts the behavior of the social group from
being cooperative and consensus seeking to being antagonistic and disagreeable. Once in this region of T ≥ 1
and S ≤ 0, the domain of the PD game, the fraction
of protagonists can be greater than that of the isolated
EGM (Fig. 5d) and still not produce a phase transition
and therefore the group does not reach consensus.
The DMM-EGM network dynamics is depicted using
the global variable ξ(K, t) for a subcritical and a supercritical value of the control parameter at selected values
of the sucker payoff S and temptation T in Figure 7a. In
the subcritical case the DMM-EGM network dynamics
are seen to be random. The global variable ξ(K, t) time
series have large scale fluctuations for T < 0 and S > 0,
with much more rapid large amplitude fluctuations in
the positive temptation regions. In the supercritical region the DMM-EGM network transitions to consensus for
T < 0 are characterized by a very low amplitude fluctuations once consensus is achieved. However in the other
ST regions the value of the control parameter seems to
be irrelevant and large amplitude fluctuations of ξ(K, t)
are observed.
Another way to view the dynamics of the coupled
DMM-EGM network is to record the difference in the
number of changes inflicted by the DMM layer on the
EGM layer and the number of changes inflicted by the
EGM layer on the DMM layer at each time step. Thus,
it is the difference in the number of changes performed
at the second and fourth step of the simulation. The
time series recorded in Figure 7b have the same parameter values as those depicted in Figure 7a. First we
notice the insensitivity of the behavior of the time series to the K value of the dynamics. Next is the balance in the two kinds of change apparent for negative
temptation T < 0 with relatively low amplitude fluctuations. Whether there are more changes projected from
the DMM or EGM layer to the complementary layer
varies with the S and T values. For example at T = 3.0
FIG. 6: (Color online) Scaling of the global order parameter (top left panel, middle row) and the susceptibility (top
right panel, bottom row) in the vicinity of the phase transition point. Top panel illustrates the scaling present in an isolated DMM layer with critical control parameter KC = 1.70.
In middle and bottom rows, the columns correspond to the
dual-layer configuration with T = 0, S = 2, T = 0.5, S = 0.5
and T = −0.5, S = −0.5 for the left, middle and right column respectively. Lattice size is N = 20 x 20, and scaling
exponents are β = 1/8 and γ = −7/4.
the two changes are equal for S = 2.5 and DMM exceeds
EGM changes at S = 1.0, with relatively large fluctuations in both cases. This suggests that four regions of
dynamics of DMM-EGM network, identified on Fig. 5
are characterized by the balance between the DMM and
EGM layer in the case that presents phase transition, and
an imbalance between the two mechanisms of decisionmaking in other cases.
V.
CONCLUSION
We expect that the behavior present in the dual layer
model is not limited to the regular lattice configuration
discussed herein. The DMM demonstrates the cooperative behavior in a wide range of network topologies, where
the phase transition is observed for both random, smallworld and scale-free configurations [29]. Similarly, the
cooperation based on the Prisoner’s Dilemma is present
in heterogenous topologies [30]. Additionally, recent experimental studies of the cooperation when humans play
a Prisoner’s Dilemma demonstrate that both the regular
7
FIG. 7: (Color online) a) Here the global variable ξ (K, t) is depicted as a function of time for two values of K and a sequence
of ST values. b) Plotted is the difference between the fraction of nodes whose strategy was changed from the EGM to the
DMM layer minus that changed from the DMM to the EGM layer at each time of the simulation.
lattice and a scale-free network reach the same level of
cooperation, which is comparable with the level of cooperation of smaller homogeneous networks [31, 32].
In this paper we considered two dynamically coupled
networks; the dynamics of one being determined by imitation using the DMM [25] and the other following the
game theoretic format prescribed by Nowak and May
[14]. We find that in the domain T ≤ 1, R = 1, P = 0 and
all S the protagonists asymptotically dominate and phase
transition behavior is robust. Critical dynamics occurs
in this region even when the initial fraction of antagonists would be sufficient to inhibit critical behavior for
the DMM network in isolation. Consequently the EGM
layer increases the domain of initial states attracted to
critical dynamics for the two-layer network dynamics.
On the other hand, the influence of the EGM antagonists on the DMM-EGM network dynamics is quite dif-
ferent from that of committed minorities [27, 28], even
when the committed minorities are modeled as a distinct
dynamic network [25]. The EGM layer can actually increase the stability of the consensus-making process for
the two-layer network model. Consequently, consensus
can be facilitated by payoffs even in cases where intuition
might dictate that antagonists would prevail. This suggests the possibility of counter-intuitive policies, which
society might adopt, that could facilitate the consensusmaking of large groups, even in the face of what might
appear to be overwhelming polarization.
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VI.
ACKNOWLEDGEMENT
The authors would like to thank the Army Research
Office for support of this research.
8
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